Three circles of radius r touching each other are inscribed inside a circle.…

Concept

Three circles of equal radius r that touch each other pairwise have their centres exactly r + r = 2r apart, so the three centres form an equilateral triangle of side 2r. For an equilateral triangle of side a, the circumradius (centroid-to-vertex distance) equals a/√3. The circle that just encloses all three small circles is centred at this same centroid, and its radius equals that circumradius plus the small circle's own radius r — the outer circle must reach one more r past each centre to just touch the small circle from outside.

Application

1. Each pair of circles touches, so the centre-to-centre distance = r + r = 2r. The triangle formed by the three centres is therefore equilateral with side a = 2r.

2. The circumradius of this centre-triangle equals side/√3 = 2r/√3 ≈ 1.1547r.

3. The outer circle's radius equals this circumradius plus the small circle's own radius r: R = 2r/√3 + r.

4. R = r(1 + 2/√3) = r(1 + 1.1547) ≈ 2.155r.

Cross-check

Using the centroid-to-vertex distance via cos30°: distance = r / cos30° = r / (√3/2) = 2r/√3 — the same 1.1547r, confirming the outer radius is ≈ 2.155r.